Problem: Determine how many solutions exist for the system of equations. ${2x+y = 3}$ ${-2x-y = 5}$
Solution: Convert both equations to slope-intercept form: ${2x+y = 3}$ $2x{-2x} + y = 3{-2x}$ $y = 3-2x$ ${y = -2x+3}$ ${-2x-y = 5}$ $-2x{+2x} - y = 5{+2x}$ $-y = 5+2x$ $y = -5-2x$ ${y = -2x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x+3}$ ${y = -2x-5}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.